Gelfand--Kirillov dimensions of highest weight modules for basic classical Lie superalgebras
Abstract
We give exact Gelfand--Kirillov (GK) dimension formulas for simple highest weight modules of the basic classical Lie superalgebras considered in this paper. For the type II orthosymplectic algebras D(m,n) and B(m,n), we identify the PBW associated variety as a product of factor supports evaluated at highest weights for two compatible Borel subalgebras. This gives algorithms for arbitrary complex highest weights, without integrality or typicality hypotheses. In type B, the terminal factor is osp(1|2n), for which we derive an insertion formula that distinguishes integer and half-integer shifted coordinates. For G(3), F(4) and D(2,1;α), finite odd-reflection procedures give exact formulas, their GK spectra, and finite-dimensionality criteria; we also describe the dependence on α in the last case. In type I, the GK dimension equals that of the simple even highest weight module for the distinguished Borel. The computations use the established ordinary Lie algebra formulas involving Lusztig's a-function and Robinson--Schensted insertion.
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BibTeX
@misc{jiang2026gelfandkirillov,
title = {{Gelfand--Kirillov dimensions of highest weight modules for basic classical Lie superalgebras}},
author = {Jing Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.11551},
eprint = {2606.11551},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.11551},
}
RIS
TY - UNPB
TI - Gelfand--Kirillov dimensions of highest weight modules for basic classical Lie superalgebras
AU - Jing Jiang
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2606.11551
UR - https://arxiv.org/abs/2606.11551
AB - We give exact Gelfand--Kirillov (GK) dimension formulas for simple highest weight modules of the basic classical Lie superalgebras considered in this paper. For the type II orthosymplectic algebras D(m,n) and B(m,n), we identify the PBW associated variety as a product of factor supports evaluated at highest weights for two compatible Borel subalgebras. This gives algorithms for arbitrary complex highest weights, without integrality or typicality hypotheses. In type B, the terminal factor is osp(1|2n), for which we derive an insertion formula that distinguishes integer and half-integer shifted coordinates. For G(3), F(4) and D(2,1;α), finite odd-reflection procedures give exact formulas, their GK spectra, and finite-dimensionality criteria; we also describe the dependence on α in the last case. In type I, the GK dimension equals that of the simple even highest weight module for the distinguished Borel. The computations use the established ordinary Lie algebra formulas involving Lusztig's a-function and Robinson--Schensted insertion.
ER -